## Abstract

The number of rank two ℚ_{l}-local systems, or ℚ_{l}-smooth sheaves, on (X — {u}) ⊗𝔽_{q} 𝔽, where X is a smooth projective absolutely irreducible curve over 𝔽_{q}, 𝔽 an algebraic closure of 𝔽_{q} and u is a closed point of X, with principal unipotent monodromy at u, and fixed by Gal(𝔽/𝔽_{q}), is computed. It is expressed as the trace of the Frobenius on the virtual ℚ_{l}-smooth sheaf found in the author’s work with Deligne on the moduli stack of curves with etale divisors of degree M ≥ 1. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group π_{1}((X — {u}) ⊗_{𝔽q } 𝔽) invariant under the Frobenius Fr_{q} with principal unipotent monodromy at u, or cuspidal representations of GL(2) over the function field 𝔽 = 𝔽_{q}(X) of X over 𝔽_{q} with Steinberg component twisted by an unramified character at u and unramified elsewhere, trivial at the fixed idèle α of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at f_{u}Π_{v≠u}χK_{v}, with an Iwahori component f_{u} = χ_{Iu}/\Iu\, hence also the pseudo-coefficient χI_{u}/\Iu\ — 2 χK_{u} of the Steinberg representation twisted by any unramified character, at u. Theorem 2.1 records the trace formula for GL(2) over the function field F. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of ℚ_{l}-local systems, or ℚ_{l}-smooth sheaves, on X ⊗_{Fq } 𝔽, fixed by Fr_{q}, namely ℚ_{l}-representations of the absolute fundamental group π(X ⊗_{𝔽q } 𝔽) invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of GL(2) trivial at a fixed idele α of degree 1. This number is expressed as the trace of the Frobenius of a virtual ℚ_{l}-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function Π_{v} χK_{V} of the maximal compact subgroup, with volume normalized by |K_{v}| = 1. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on X and deg(S), and not on the degrees of the points in S_{1}.

Original language | English |
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Pages (from-to) | 739-763 |

Number of pages | 25 |

Journal | American Journal of Mathematics |

Volume | 137 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2015 |